Spectral analysis and geometry of a sub-Riemannian structure on [formula omitted] and [formula omitted

Abstract

The purpose of this paper is to study the spectral properties of a sub-Laplacian on S 3 , i.e., we discuss the analytic continuation of its spectral zeta function, give explicit expressions of the residues and especially, we provide an expression of the zeta-regularized determinant of the sub-Laplacian on S 3 . Also, we describe sub-Riemannian curves on S 3 based on the Hopf bundle structure, together with a proof of Chow’s theorem for this case in a strong sense (= connecting property by globally smooth curves). A characterization of sub-Riemannian geodesics on S 3 via an isoperimetric problem through the Hopf bundle is explained. Incidentally, we introduce a hypo-elliptic operator on P 1 C descended from the sub-Laplacian on S 3 , which we call a spherical Grushin operator. We determine the subspace where it degenerates and give an expression of the trace of its heat kernel by making use of the trace of the heat kernel of the sub-Laplacian. In case of S 7 , we limit ourselves to present the spectral zeta function of a sub-Laplacian.